


Elimination of Dummy Variables

by Bitenomnom



Series: Mathematical Proof [20]
Category: Sherlock (TV)
Genre: A Study in Pink, Alternate Universe, Gen, apparently in a Sherlockless life John ends up getting his kicks elsewhere, dark!john
Language: English
Status: Completed
Published: 2012-10-09
Updated: 2012-10-09
Packaged: 2017-11-15 23:14:15
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,994
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/532836
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>Sherlock's after another serial killer, and so soon after the case with the cabbie, it's another Christmas. This one's ex-military, with strong moral principles -- and a particular type of victim. Sherlock can't seem to catch him -- or maybe it's just that he <i>won't</i>.</p>
            </blockquote>





	Elimination of Dummy Variables

**Author's Note:**

> Another "not sure how much the math is related" one. Again, didn't have a ton to work with today. But, I liked the idea that came to mind, so I went with it. XD
> 
> I really wish I would've had more time to develop and write this one. Perhaps one day in the distant future I can rehash it with a little more detail and thought/planning. (For instance, there was a lot more I wanted to do with John, so I think his development didn't come through as well as I'd have liked/as I'd have done it if I had more time...)

When setting up a two-way analysis of variance problem (see [Parallel](http://archiveofourown.org/works/528616)), a natural way to account for all of the variables is to use a dummy variable for each possible row, column, and combination thereof. However, it has previously been established (see [Between Variation](http://archiveofourown.org/works/526920)) that for a problem of n categories, n-1 dummy variables can be used to describe it; simply using a dummy variable for every combination results in far too many. However, it is possible to eliminate many of these by writing them as a combination of other variables. For instance, in doing the two-way ANOVA we decide that the sum of the estimators for each row and column is 0, so if you have two columns, then for one particular row, the value for one column must be equal and opposite to the other.

 

We also need to have a number of equations appropriate to the number of parameters, which is why such restrictions as “the sum of the estimators for each row and column is 0” are added. If we have more parameters than equations, the equations won’t be linearly independent and we have infinitely many solutions. If we have more equations than parameters, we won’t be able to find an answer (unless some of the equations aren’t linearly independent from one another).

 

 

 

***

 

 

 

            Sherlock couldn’t catch him.

            Or he _wouldn’t_ catch him.

            This was just good enough that he didn’t know the difference. It was Christmas; it was Mycroft’s dissatisfied sneer. Two serial murderers back to back and one he couldn’t catch, and more, so much more, that he had unearthed and overturned: spiderwebs laid to _waste_ with the power of his mind alone.

 

 

            There was a reason Sherlock, if he could catch him, if he did catch him, _wouldn’t_.

            It was his victims.

            At first, there seemed to be no connection: an actuary, a horse trainer, an old woman, a young man. Killed in completely different places, at completely different times, with no trace left behind besides bullets from the same weapon. Different family, different friends, different previous jobs, different previous schools and universities and social groups. Some had enemies. Some didn’t.

            When Sherlock began to see the connection, he burst into giggling. Mrs. Hudson (bless her, still under the impression he was looking for a _flatmate_ , of all things) informed him it wasn’t decent. He didn’t care, oh, he didn’t care. This was better than Jim, Jim who turned out to be only a stop on the route to _this_. Jim had been all good and well. He was clever, yes—possibly, though there was no way of telling, yet, cleverer than this murderer—but he was also _stupid_. Oh, his games had been fun. But he had stepped out and played; he had all but forced himself into Sherlock’s life. Jim played chess, and it was a good game—but a game with an inevitable end, and a winner, and a loser. This one played _war_.

 

 

 

            This one played war on a particular type of person.

            _Bad_ people. _Stupid_ people. The ignorant, the undeserving.

            Mitchell Tipton, 37, surreptitiously stole small but frequent amounts from donation collections meant to go to hospitals. He was stupid enough that he had saved files detailing the amounts on his very unsecure mobile.

            Laryn Peck, 22, threw bottles at homeless people and had an IQ of 72.

            Zachary Coburn, 54, managed a reasonably large business firm. He was killed one week after the _Evening Standard_ printed a short article spotlighting his superb treatment of his employees. This one had stumped Sherlock until he dug further and realized that in the twenty-five years since the start of his firm, he had never hired a woman.

            On the list went: a money launderer, a thief, a rapist, a doctor who cheated on his exams, a minor politician who played a little too underhanded. No wonder Mycroft grimaced. But he could take care of himself.

            Sherlock wondered if the average IQ of London had risen since this murderer had gotten started.

 

 

 

            What Sherlock never would have suspected, at least not until London had lost ten or so of its least desirable occupants, was that he had already encountered the murderer once. He _should_ have made the connection straight away. The first victim was too distinct from the others—too clever. Bad, yes, very bad, but not the sort of bloke the murderer would have had high on his priority list; and anyway, Sherlock caught him first, and nobody else had figured it out before him. Nobody else had figured out about _the cabbie._

            He had concluded that whoever had shot the cabbie was military, and so he had briefly hunted down records of recent discharges—probably dishonorable. They led him to one Sebastian Moran, who led him to one James Moriarty. But neither were _this_ man. Jim had played his game and it was fun, but felt, to Sherlock, like playing poker without chips, as if Jim had expected to find _something_ to play against Sherlock but hadn’t. Of course, he was in prison, now—in Mycroft’s custody, really. Perhaps Sherlock should’ve taken a leaf out of the murderer’s book and killed Jim—but it seemed such a waste. Dismantling Jim’s web had granted him some satisfaction, required some serious brainpower, had, at the time, been utterly pleasing to tear to pieces, but it took less than two months, and it faded in the face of this case.

 

 

 

            Lestrade was required to want to catch him, on paper anyway. “He _is_ saving us a lot of trouble, though the killing’s a bit over the top,” he said to Sherlock. “I mean, even if the victims are awful, they would’ve just gone to prison. Shooting ‘em in the head isn’t exactly a fair legal procedure. Some of ‘em weren’t all _that_ bad.”

            “True enough.” Those were details Lestrade had to worry about, but not Sherlock. If Sherlock caught him, he wondered, would he turn him in?

 

 

 

            The connection should have been obvious.

            Sherlock had concluded military. If he’d only spent a moment more _thinking_ he’d have realized that whether the cabbie was cleverer than the victims that followed or not, he was still _not good_ , and the person who shot him took the shot the instant he was about to kill again—proof he would do it, proof he’d go through with it. Morals, morals. The serial killer and the shooter, _his_ shooter, whoever had shot the cabbie before Sherlock could take the pill—both had them. Sherlock found the ballistics report: same weapon.

 

 

 

            When he caught him, it was completely by accident.

            It was a stakeout: if the brother had stolen his mother’s jewelry, he would be sneaking out this way, through this alley. He’d be coming through any time, Sherlock suspected, based on the habits he’d observed. When he heard the boy coming, he would pretend to be disoriented and lost, jostle him a little, capture a reaction and use that to gauge his guilt. If he was guilty: Sherlock would tackle him to the ground. It was never the safest approach, but he had few other options.

            Sherlock pressed himself against the wall and peeked around the corner. No one coming—yet. With his nose still poked around the corner, he heard from behind him the light _click_ of a gun being cocked.

            He froze.

            Almost inaudible breathing was the only further indication of a presence behind him, that and the faint rustle of shifting weight.

            Sherlock half expected to find the gun at his temple or his back, expected demands for his wallet or whatever else he had on him. Or maybe this was retribution: someone he’d put in prison had concerned family.

            When there were no words, when he felt no gun pressed against him, he turned his head back to look over the man holding it.

            Stance: military. Sharp, sure eyes: definitely in this very spot for a specific reason. Very slight difference in the way he held his leg when he shifted his weight again: previous injury. No—because it stopped once he finished shifting his weight. Psychosomatic limp, but not bad enough that he carried a cane. Gun held—away from the both of them, hand carefully wrapped around it—in perfectly steady hands. Ex-military, and _missing_ it. Then: discharged from the military due to injury, something other than the leg, unable to go back.

            Strong, steely stare. Flared nostrils, calm rising and falling of a reasonably developed chest, steady positioning the result of well-directed adrenaline. Hair: lightly mussed, indiscernible in color in the dark of night. His gaze pinned Sherlock to the wall, pierced him with something hotter than a freshly fired bullet. Different than what he’d imagined: he’d thought icy, perhaps, cool like pooled metal. But then: he’d also thought tall, spidery, not solid and compact. He was thinking red giant and here was a blue dwarf, not cold and looming but unassuming and burning, burning, _burning_. “It’s you,” Sherlock finally breathed. Because it was.

            “I could say the same,” he answered. His whisper: so evenly blended with the night that Sherlock wasn’t sure he’d heard it, but for that he’d been waiting to hear it for months and months and now he would not forget it.

            “Sherlock Holmes,” Sherlock said, raising an eyebrow expectantly, waiting for the spark of panic or recognition or whatever it would yield. And a name—a _name_. _His_ name.

            “I know,” was all he said.

            “So you know I should call Scotland Yard this instant.”

            “I know.”

            “And you don’t care?”

            “Mister Holmes—”

            “Sherlock.”

            “Sherlock, I have killed a total of twenty-three people in my life, twelve since I came back from Afghanistan. I am a bloody awful person. Call the coppers. I don’t care.”

            “You saved me.” _Technically_ he did. _Technically_ Sherlock would have taken the pill.He _could_ have died.

            “They were both poisoned. You know that, right?”

            He had considered the possibility. “Yes.”

            “So are you just suicidal, or what?”

            “Says the serial killer who just asked me to turn him in.” Sherlock paused. “Why?”

            “Why ask you to turn me in? Again, because I’m a s—”

            “Why do it at all?”

            The man let out a long breath and pressed one hand against his leg. “Let’s call it a long story and save it. I’ll write you letters from prison about it.”

            One corner of Sherlock’s mouth turned up. “What’s your name?”

            “John Watson,” John held out the hand not holding the gun to shake Sherlock’s. As if suddenly aware of its presence, he flipped the safety back on and tucked it in the back of his trousers.

            “Changed your mind, John?” Sherlock nodded toward the stowed gun.

            “Well, you’re taking me in, aren’t you?”

            “Mm,” was all Sherlock said. He pressed his fingers together and rested them in front if his mouth for a few moments. “I believe I can summarize your story for you. You have a psychosomatic limp, but it doesn’t bother you as much now as it used to—just now, thinking about the past, you were wincing slightly and rubbing at your leg. It resulted from something that happened while you were in the army, and either the event triggering it occurred at the same time as the injury that lead to your discharge, or it occurred prior to it but the limp didn’t bother you at the time. Either way, you clearly forget about it in high-adrenaline situations. You live alone—who else could get away with what you’re doing unless living with an exceptionally dull flatmate, which you would never stand for given the profiles of your victims. Therefore: no close friends or family with whom to share lodgings since your return several months ago. So: alone, previously traumatized, and an adrenaline junkie.

“You returned to London, John Watson, and became _miserable._ Therapist didn’t help in the slightest. Your limp, which you may not have been completely aware of while in Afghanistan or while recovering, came to you at full force, and you discovered that the only thing that cured it was the sort of rush that didn’t come with your boring, ordinary life. You stumbled across the cabbie—perhaps you were in the area and saw me being escorted into the building at gunpoint, and took up post at the adjacent building. You saw what was about to happen, concluded correctly that this was the killer of the other ‘suicide’ victims, and shot.

“And it did the trick, and you felt fantastic about it. And yet: now you had murdered outside the battlefield, outside orders. But the world, you thought, was a better place for it, and based on your stance and the way you immediately turned yourself in, you, John Watson, are in for a penny, in for a pound. So you continued.” He stared at John, obviously awaiting a reaction. “Well?”

            “Amazing,” John breathed. “That was—amazing. Brilliant.”

            “Was I right?”

            “Yes,” he said. “God, yes.”

            “About _everything_?”  
            John crossed his arms. “I did see you being led in at gunpoint. Of course, I also saw later that it wasn’t actually a gun—but it was tough to tell in the dark, of course, when he led you in. When I saw it was a cabbie doing it, and when I saw him get out the little bottles, I thought of the suicides that couldn’t possibly be suicides they’d been talking about on the telly, and I thought, _oh_ , a cab driver—of course. No wonder the police didn’t catch on. I waited to make sure, and then, yeah—I shot him.”

            “But…?”

            “But it wasn’t just that somebody was going to die. Christ, it’s not as if _shoot him, shoot him_ is the first thing that comes to my head for everything. But I recognized you. My friend Mike Stamford told me about your site. I thought _that’s him, that’s Sherlock Holmes, he’s only solved about fifty crimes this winter, that brilliant nutter who catalogues tobacco ash,_ and I had to do it. I had to save you.”

            “You shot because it was _me_?”

            John glanced at his feet. “I s’pose so, yeah.”

            “And that’s what led you to continue on. So it’s because of _me_.”

            “I wouldn’t—”

            “John, I propose an alternate solution to turning you in.” Because he hadn’t wanted to from the start, really, and he especially didn’t want to _now_. John was interesting. John was more than that—obviously very intelligent, and other—other _things_ , physically capable, could probably take or dole out a beating as well as anyone, probably breathed ragged breaths but continued on with the tenacity of a pitbull—but he was interesting, and that, surely, was the most important. It was the perfect solution.

            “Look, just because you feel guilty for starting me on this path doesn’t mean you should—”

            “I don’t feel guilty,” Sherlock interrupted. “Ever.”

            “Mm,” John’s lips pressed together. “I see.”

            “I have a way to keep you from killing again, but also keep you from being miserable, if you are amenable to the idea.”

            John waited.

            “You work with me instead. On the other end of the homicides, as it were.”

            “You think that would help?”

            “It’s more exciting than killing people. I thought it might appeal to your sense of morality—atoning for your sins by helping the police, or whatever.”

            “I doubt they’ll let me.”

            “Oh, of course they will.” John raised an eyebrow. “They’ll wonder where this John Watson bloke came from, and then they’ll start wondering what happened to that serial killer from a couple months ago, but they needn’t conclude there is a connection.”

            “Sherlock…”

            “Be my flatmate, John. Be my assistant. And if you decide you need to turn yourself in, I have Detective Inspector Lestrade on speed-dial on my mobile.”

            John took his steady, even breaths. His fists balled and unballed. “What about this guy, then?” he nodded toward the street Sherlock had been watching.

            “Oh, leave him,” Sherlock shrugged. “We’ll find a different way to prove he’s guilty tomorrow.”

            “Right,” John said with one determined nod, followed, after a few seconds, by a hint of a smile creeping across his jaw and into his lips.

            “Is that a yes?” Sherlock tried to keep it even, calm, cool, but John’s smile was infectious, and from the gleam that burst into John’s eyes Sherlock was certain his were eager.

            “I hardly know anything about you.”

            “I don’t think _you_ , of all people, have a lot of room to be picky about flatmates, John,” Sherlock smirked, and couldn’t hold back a slight chuckle.

            John snickered in return. “Sherlock, we can’t _giggle_ about my being a murderer.”

            Sherlock seemed doubtful, and only continued to grin at John.

            “Okay, fine, yes, let’s give it a try,” John breathed, leaning back against the wall.

            Sherlock wanted to clap a hand to his shoulder, or whirl him around, or perhaps wrap his arms around him, but he’d do none of those things—not now, anyway. Not yet. “Hungry?” he asked instead.

            “Starving.” On the balls of his feet: ready to go. Ready to leave. Flexible, not at all stuck, ready to be pulled out of the dark. Ready to be scrubbed off and to start over, if such a thing would be possible for such a man, all covered in memories and scars. Ready, just _ready_ , and Sherlock got the impression that that was, would be, _John_ , always, whether he came out clean or whether the whiff of blood couldn’t be scrubbed out, he’d be ready. Ready for Sherlock. _Ready_.  
            “I know a good Chinese that stays open until two a.m.” Sherlock took off down the street, leaving John to follow and then match his pace. They fell into step naturally; John walked like heavy layers were floating off behind him, left abandoned on the sidewalk. “You can always tell a good Chinese by the bottom third of the door handle…”


End file.
